Question

The following system of equations

x₁ + x₂ + 2x₃ = 1

x₁ + 2x₂ + 3x₃ = 2

x₁ + 4x₂ + ax₃ = 4

has a unique solution. The only possible value(s) for a is/are 

Answer 1

Correct Answer - Option 4 : any real number other than 5.

For unique solution

Rank (A|B) = Rank (A) = No. of unknowns

For No solution

Rank (A|B) = Rank(A) < No.of unknows

For No solution

Rank (A|B) ≠ Rank (A)

Calculation:

Now the augmented matrix A|B is

\(\left[ {\begin{array}{*{20}{c}} 1&1&2&1\\ 1&2&3&2\\ 1&4&a&4 \end{array}} \right]\) 

Performing R_2­ → R₂ – R₁and R₃ → R₃ – R₁we get

\(\left[ {\begin{array}{*{20}{c}} 1&1&2&1\\ 0&1&1&1\\ 0&3&{a - 2}&3 \end{array}} \right]\) 

Now, R₃ → R₃ – 3R₂

\(\left[ {\begin{array}{*{20}{c}} 1&1&2&1\\ 0&1&1&1\\ 0&0&{a - 5}&0 \end{array}} \right]\) 

Now for unique solution, Rank(A) = Rank (A|B)

A – 5 ≠ 0

⇒ a can take any real value except 5.